Summary remarks on momentum theory

 Summary remarks on momentum theory

The place of momentum theory is that it gives a broad understanding of the functioning of the rotor and provides basic relationships for the induced velocity created and the power required in producing a thrust to support the helicopter. The actuator disc concept, upon which the theory is based, is most obviously fitted to flight conditions at right angles to its plane, that is to say the hover and vertical flight states we have discussed. Nevertheless further reference to the theory will be made when discussing forward flight (Chapter 5). Momentum theory brings out the importance of disc loading as a gross parameter: it cannot however look into the detail of how the thrust is produced by the rotating blades and what design criteria are to be applied to them. For such information we need additionally a blade element theory, corresponding to aerofoil theory in fixed-wing aerodynamics: to this we shall turn in Chapter

Complete induced-velocity curve

 Complete induced-velocity curve

It is of interest to know how the induced velocity varies through all the phases of axial flight. For the vortex-ring and turbulent-wake states, where momentum theory fails, information has been obtained from measurements in flight, supported by wind tunnel tests (Gustafson (1945), Gessow (1948), Brotherhood (1949), Castles and Gray (1951) and others). Obviously the making of flight tests (measuring essentially the rate of descent and control angles) is both difficult and hazardous, especially where the vortex-ring state is prominent, and not surprisingly the results show some variation: nevertheless the main trend has been ascertained and what is effectively a universal induced-velocity curve can be defined. This is shown in Fig. 2.10, using the simple momentum-theory results of Equations (2.10) and (2.13) in the regions to which they apply. We see that moving from hover into descent the induced velocity increases more rapidly than momentum theory would indicate. The value rises, in the vortex-ring state, to about twice the hover value, then falls steeply to about the hover value at entry to the windmill-brake state.

Vertical descent

 In vertical descent the nature of flow through the rotor undergoes significant changes. The stream velocity Vc is now negative while the induced velocity vi remains positive as the rotor continues to maintain lift. Initially small recirculating regions develop around the blade tips, as shown in Fig. 2.5. Becoming evident when Vc reaches a level about half vi, an interaction takes place between the upward flow around the disc and the downward flow through it, resulting in the formation of a vortex ring encircling the rim of the disc, doughnut fashion.The situation is illustrated in Fig.2.6.As this vortex-ring state develops the flow becomes very unsteady and the rotor exhibits high levels of vibration. It appears that the ring vortex builds up strength and periodically breaks away from the disc, spilling haphazardly into the flow and causing fluctuations in lift and also in helicopter pitch

and roll. Flight in the developed vortex-ring state, which reaches its worst condition when the descent rate is about three quarters of the hover induced velocity, is unpleasant and potentially dangerous. Because of the dissipation of energy in the unsteady flow, simple momentum theory cannot be applied. As the descent rate approaches the level of the induced velocity, a modified state is observed in which, corresponding to the near equality,there is little or no net flow through the disc.Now the flow is characterized by vortices shed into the wake in the manner of the flow around a solid bluff body. In this turbulent-wake state (Fig. 2.7) flight is still rough but less so than in the vortex-ring state. Simple momentum theory is again not applicable, since energy is dissipated in the eddies of the wake. At large descent rates, when Vc is numerically greater than about 2vi, the flow is everywhere upwards relative to the rotor, producing a windmill-brake state, in which power is transferred from the air to the rotor. With a flow pattern as in Fig. 2.8, simple momentum theory gives a reasonable approximation: thus with Vc negative and vi positive the thrust is:

Momentum theory for hover

The helicopter rotor produces an upward thrust by driving a column of air downwards through the rotor plane. A relationship between the thrust produced and the velocity communicated to the air can be obtained by the application of Newtonian mechanics – the laws of conservation of mass,

momentum and energy – to the overall process. This approach is commonly referred to as the momentum theory for helicopters.It corresponds essentially to the theory set out by Glauert1 for aircraft propellers, based on earlier work by Rankine and Froude for marine propellers. The rotor is conceived as an ‘actuator disc’, across which there is a sudden increase of pressure, uniformly spread. In hover the column of air passing through the disc is a clearly defined streamtube above and below the disc: outside this streamtube the air is undisturbed. No rotation is imparted to the flow.

The situation is illustrated in Figs 2.1a–2.1c. As air is sucked into the disc from above, the pressure falls. An increase of pressure Dp occurs at the disc, after which the pressure falls again in the outflow, eventually arriving back at the initial or atmospheric level p•. Velocity in the streamtube increases from zero at ‘upstream infinity’ to a value vi at the disc and continues to increase as pressure falls in the outflow, reaching a value v• at ‘downstream infinity’. Continuity of mass flow in the streamtube requires that the velocity is continuous through the disc. Energy conservation, in the form of Bernoulli’s equation, can be applied separately to the flows before and after the disc. Using the assumption of incompressible flow, we have in the inflow:

WING DIHEDRAL FOR LATERAL STABILITY

The most common method of obtaining lateral stability is by the use of a dihedral angle on the main planes (Figures 4.1 ). Dihedral angle is taken as being the angle between each plane and the horizontal, not the total angle between the two planes, which is really the geometrical meaning of dihedral angle. If the planes are inclined upwards towards the wing tips, the dihedral is positive; if downwards, it is negative and called anhedral (Figure 4.2); but the latter arrangement is only used in practice for reasons other than stability, or even to reduce the amount of lateral stability.

METHODS OF LATERAL STABILITY

METHODS OF LATERAL STABILITY

From what has been said earlier, an aircraft has lateral stability if, following a roll displacement, a restoring moment is produced which opposes the roll and returns the aircraft to a wings level position. In that, aerodynamic coupling produces rolling moments that can set up side slip or yawing motion. It is therefore necessary to consider these interactions when designing an aircraft to be inherently statically stable in roll. The main contributors to lateral static stability are:

• Wing dihedral,
• Sweepback,
• High wing position,
• Keel surface.

A design feature that has the opposite effect to those given above, i.e. that reduces stability is anhedral. The need to reduce lateral stability may seem strange, but combat aircraft and much high speed automatically controlled aircraft, use anhedral to provide more maneuverability.

METHODS OF LATERAL STABILITY

METHODS OF LATERAL STABILITY

Lateral and directional stability are so closely interconnected that it is impossible to consider one without the other, and they are therefore often grouped together under the single term of lateral stability. But, for simplicity, we will first consider them separately; then we, shall try to see how they affect each other.

Methods: From what has been said earlier, an aircraft has lateral stability if, following a roll displacement, a restoring moment is produced which opposes the roll and returns the aircraft to a wings level position. In that, aerodynamic coupling produces rolling moments that can set up side slip or yawing motion. It is therefore necessary to consider these interactions when designing an aircraft to be inherently statically stable in roll. The main contributors to lateral static stability are:

• Wing dihedral,
• Sweepback,
• High wing position,
• Keel surface.

A design feature that has the opposite effect to those given above, i.e. that reduces stability is anhedral. The need to reduce lateral stability may seem strange, but combat aircraft and much high speed automatically controlled aircraft, use anhedral to provide more maneuverability.
3.3.3. Wing dihedral and lateral stability: Dihedral angle is defined as the upward inclination of the wings from the horizontal. The amount of dihedral angle being dependent on aircraft type and wing configuration i.e. whether the wings are positioned high or low with respect to the fuselage and whether or not they are straight or swept back.

The righting effect from a roll using wing dihedral angle may be considered as a two stage process; where the rolling motion is first stopped and then the down-going wing is returned to the horizontal position.
So we first stop the roll. In Figure 3.6(a) we see that for an aircraft in a roll, one wing will move down and the other will move up, as a result of the rolling motion. The vector diagrams (Figure 3.6(b)) show the velocity resultants for the up-going and down-going wings. The direction of the free stream airflow approaching the wing is changed and the AOA on the down going wing is increased, while the AOA on the up-going wing is decreased (Figure 3.6(c)). This causes a larger CL and lift force to be produced on the lower wing and a smaller lift forces on the upper wing, so the roll is stopped. When the roll stops the lift forces equalize again and the restoring effect is lost. In order to return the aircraft to the equilibrium position, dihedral angle is necessary. A natural consequence of banking the aircraft is to produce a component

In Figure 3.7(a) the component of lift resulting from the angle of bank can clearly be seen. It is this force that is responsible for sideslip. Now if the wings were straight the aircraft would continue to sideslip, but if dihedral angle is built-in the sideways air stream will create a greater lift force on the down-going wing (Figure 3.7(b)). This difference in lift force will restore the aircraft until it is no longer banked over and side-slipping stops.

METHODS OF AIRCRAFT STABILITY

Meaning, dynamics and different states/types of stability along with their conditions have been highlighted in previous weeks. Subsequent weeks will discuss different methods that employed in the designs of aircraft contributing to the stability.

METHODS OF LONGITUDINAL STABILITY

Conditions of longitudinal stability have been stated earlier and it was indicated that restoration moment comes from the tail plane. To have such requirement, method of setting longitudinal dihedral is the most contributing means.

Longitudinal Dihedral: The tail plane is usually set at an angle less than that of the main planes, the angle between the chord of the tail plane and the chord of the main planes being known as the longitudinal dihedral (Figure 3.1). This longitudinal dihedral is a practical characteristic of most types of aeroplane, but so many considerations enter into the problem that it cannot be said that an aeroplane which does not possess this feature is necessarily unstable longitudinally. In any case, it is the actual angle at which the tail plane strikes the airflow which matters; therefore we must not forget the downwash from the main planes. This downwash, if the tail plane is in the stream, will cause the actual angle of attack to be less than the angle at which the tail plane is set (Figure 3.2). For this reason, even if the tail plane is set at the same angle as the main planes, there will in effect be a longitudinal dihedral angle, and this may help the aeroplane to be longitudinally stable.

Suppose an aeroplane to be flying so that the angle of attack of the main planes is 4° and the angle of attack of the tail plane is 2°; a sudden gust causes the nose to rise, inclining the longitudinal axis of the aeroplane by 1°. What will happen? The momentum of the aeroplane will cause it temporarily to continue moving practically in its original direction and at its previous speed. Therefore the angle of attack of the main planes will become nearly 5° and of the tail plane nearly 3°. The pitching moment (about the centre of gravity) of the main planes will probably have a nose-up, i.e. unstable tendency but that of the tail plane, with its long leverage about the centre of gravity will definitely have a nose-down tendency. If the restoring moment causes the tail plane is greater than the upsetting moment caused by the main plane and possibly the fuselage, then the aircraft will be stable.

This puts the whole thing in a nutshell, but unfortunately it is not quite easy to analyse the practical characteristics which will bring about such a state of affairs; however the forward position of the centre of gravity and the area and leverage of the tail plane will probably have the greatest influence. It is interesting to note that a tail plane plays much the same part, though more effectively, in providing longitudinal stability, as does reflex curvature on a wing, or sweepback with wash-out of incidence towards the tips.

When the tail plane is in front of the main planes (Figure 3.3) there will probably still be a longitudinal dihedral, which means that this front surface must have greater angle than the main planes. The latter will naturally still be at an efficient angle, such as 4°, so that the front surface may be

HOW DIRECTIONAL STABILITY IS ACHIEVED

HOW DIRECTIONAL STABILITY IS ACHIEVED

To be directionally stable, an aircraft should return to the equilibrium position after a yaw disturbance. It will do this if the centre of pressure of the aerodynamic forces exerted on the side surface of the aircraft is behind the centre of gravity. This can be ensured by the attachment of a fin at the rear of the fuselage.

If the centre of gravity is well forward in the aircraft the fin may be small, but if it is a long way back the fin may need to be very large
Many modern aircraft have long fuselages with the centre of gravity well back and therefore require a large fin area. On fast aircraft the fin becomes less efficient at high speed and this means that extra size is required.

Note: Although the diagrams show a fuselage surface only, forces which affect directional stability, are also exerted on wings, external engine nacelles etc.- It has already been mentioned that fin area and position affect lateral stability, as well as directional stability.

LATERAL STABILITY

From what has been said above an aircraft has lateral stability if, following a roll displacement, a restoring moment is produced which opposes the roll and returns the aircraft to a wings level position. In that, aerodynamic coupling produces rolling moments that can set up side slip or yawing motion. It is therefore necessary to consider these interactions when designing an aircraft to be inherently statically stable in roll.

To secure lateral stability we must so arrange things that when a slight roll takes place the forces acting on the aeroplane tend to restore it to an even keel. In all aeroplanes, when flying at a small angle of attack, there is a resistance to roll because the angle of attack, and so the lift, will increase on the down going wing, and decrease on the up-going wing. But this righting effect will only last while the aeroplane is actually rolling. It must also be emphasized that this only happens while the angle of attack is small; if the angle of attack is near the stalling angle, then the increased angle on the falling wing may cause a decrease in lift, and the decreased angle on the other side an increase; thus the new forces will tend to roll the aeroplane still further, this being the cause of auto-rotation.

DYNAMIC LONGITUDINAL STABILITY

DYNAMIC LONGITUDINAL STABILITY

Longitudinal dynamic stability consists of two basic modes, one of which you have already met, phugoid (Figure 2.6). Phugoid motion consists of long period oscillations that involve noticeable changes in pitch attitude, aircraft altitude and airspeed. The pitching rate is low and because only very small changes in AOA occur, damping is weak and sometimes negative.

The second mode involves short period motion of relatively high frequency that involves negligible changes in aircraft velocity. During this type of motion, static longitudinal stability restores the aircraft to equilibrium and the amplitude of the oscillation is reduced by the pitch damping contributed by the tail plane (horizontal stabilizer). If instability was to exist in this mode of oscillation, porpoising of the aircraft would occur and because of the relative high frequency of oscillation, the amplitude could reach dangerously high proportions with severe flight loads being imposed on the structure.

DIRECTIONAL STABILITY

As you already know, directional stability of an aircraft is its inherent (built-in) ability to recover from a disturbance in the yawing plane, i.e. about the normal axis. However, unlike longitudinal stability, it is not independent in its influence on aircraft behaviour because as a result of what is known as aerodynamic coupling, yaw displacement moments also produce roll displacement moments about the longitudi-nal axis. As a consequence of this aerodynamic coupling aircraft directional motions have an effect on lateral motions and vice versa.

The nature of these motions is yawing, rolling and sideslip, or any combination of the three.
With respect to yawing motion only, the primary influence on directional stability is provided by the fin (or vertical stabilizer). As the aircraft is disturbed from its straight and level path by the nose or tail being pushed sideways (yawed), then due to its inertia the aircraft will continue to move in the direction created by the disturbance. This will expose the keel surface to the on-coming airflow. Now the fin, acting as a vertical aerofoil, will generate a sideways lift force which tends to swing the fin back towards its original position, straightening the nose as it does so.

It is thus the powerful turning moment created by the vertical fin, due to its large area and distance from the aircraft CG, which restores the aircraft nose back to its original position (Figure 2.7). The greater the keel surface area (which includes the area of the fin) behind the CG, and the greater the moment arm, then the greater will be the directional stability of the aircraft. Knowing this, it can be seen that a forward CG is preferable to an aft CG, since it provides a longer moment arm for the fin.

We finish our study of basic aerodynamics by looking briefly at the way in which aircraft are controlled. This introduction to the subject is provided here for the sake of completeness and in order to better understand the interactions between stability and control.